Logic Programming and Artificial Neural Networks
213
Section 7.4. In particular, we consider extensions of Theorem 7.4.1 to Fittingstyle operators F P , including of course the special cases of Φ P for Kleene’s
strong three-valued logic and the corresponding operator Ψ P for Belnap’s
logic FOU R. In the next section, Section 7.7, we will consider extensions
of Section 7.5.2, or in other words we will consider approximations of local
consequence operators, including Fitting-style operators, and the Gelfond–
Lifschitz operator.
In fact, one can adopt an algebraic approach to the material presented in
this section at little extra cost, but with the benefit that the results apply
to constraint logic programs (with constraints belonging to a given semiring)
and to logic programs involving uncertainty expressed via many-valued logics,
as well as to conventional logic programs. We shall not do that, however,
as it would take us too far afield, requiring a definition of logic programs
allowing elements of an abstract set (the set C in the next definition) in clause
bodies and a corresponding new definition of Fitting-style operators. Instead,
we content ourselves with sketching the development for conventional logic
programs.
40 Nevertheless, we will present the material in full generality where
it helps, ultimately specializing to logics T . Thus, we next present one of the
main definitions we need in full generality, as follows.
7.6.1 Definition Suppose that C is a set equipped with a binary operation 8.
We say that is finitely determined or that products (relative to ) are finitely
8
8
determined in C if, for each c ∈ C, there exists a countable (possibly infinite)
collection {(R
n , E
n ) | n ∈ J } of pairs of sets R
n ⊆ C and E
n ⊆ C, where each
c
c
c
c
g
R
n is finite, such that a countable (possibly infinite) product i∈M c i in C is
c
equal to c if and only if for some n ∈ J the following statements hold.
(1) R
n ⊆ {c i | i ∈ M }.
c
(2) For all i ∈ M , c i ∈ / E
n , that is, {c i | i ∈ M } ⊆ (E
n )
co , where (E
n )
co
c
c
c
denotes the complement of the set E
n .
c
We call the elements of E
n excluded values for c, we call the elements of
c
A
n
)
co
= (E
n
allowable values for c, and in particular we call the elements of
c
c
R
n required values for c; note that, for each n ∈ J , we have R
n ⊆ A
n , so
c
c
c
that each required value is also an allowable value (but not conversely). More
generally, given c ∈ C, we call s ∈ C an excluded value for c if no product
g
g
g
i∈M c i with i∈M c i = c contains s, that is, in any product i∈M c i whose
value is equal to c, we have c i = s for no i ∈ M . We let E c denote the set of
)
co
all excluded values for c, and let A c denote the complement (E c
of E c and
call it the set of all allowable values for c. Note finally that when confusion
might otherwise result, we will superscript each of the sets introduced above
40 For full details of the sketch we present here, the reader should consult the following
papers: [Seda and Lane, 2005], [Lane and Seda, 2006], [Komendantskaya et al., 2007] and
also [Lane and Seda, 2009].
213
Section 7.4. In particular, we consider extensions of Theorem 7.4.1 to Fittingstyle operators F P , including of course the special cases of Φ P for Kleene’s
strong three-valued logic and the corresponding operator Ψ P for Belnap’s
logic FOU R. In the next section, Section 7.7, we will consider extensions
of Section 7.5.2, or in other words we will consider approximations of local
consequence operators, including Fitting-style operators, and the Gelfond–
Lifschitz operator.
In fact, one can adopt an algebraic approach to the material presented in
this section at little extra cost, but with the benefit that the results apply
to constraint logic programs (with constraints belonging to a given semiring)
and to logic programs involving uncertainty expressed via many-valued logics,
as well as to conventional logic programs. We shall not do that, however,
as it would take us too far afield, requiring a definition of logic programs
allowing elements of an abstract set (the set C in the next definition) in clause
bodies and a corresponding new definition of Fitting-style operators. Instead,
we content ourselves with sketching the development for conventional logic
programs.
40 Nevertheless, we will present the material in full generality where
it helps, ultimately specializing to logics T . Thus, we next present one of the
main definitions we need in full generality, as follows.
7.6.1 Definition Suppose that C is a set equipped with a binary operation 8.
We say that is finitely determined or that products (relative to ) are finitely
8
8
determined in C if, for each c ∈ C, there exists a countable (possibly infinite)
collection {(R
n , E
n ) | n ∈ J } of pairs of sets R
n ⊆ C and E
n ⊆ C, where each
c
c
c
c
g
R
n is finite, such that a countable (possibly infinite) product i∈M c i in C is
c
equal to c if and only if for some n ∈ J the following statements hold.
(1) R
n ⊆ {c i | i ∈ M }.
c
(2) For all i ∈ M , c i ∈ / E
n , that is, {c i | i ∈ M } ⊆ (E
n )
co , where (E
n )
co
c
c
c
denotes the complement of the set E
n .
c
We call the elements of E
n excluded values for c, we call the elements of
c
A
n
)
co
= (E
n
allowable values for c, and in particular we call the elements of
c
c
R
n required values for c; note that, for each n ∈ J , we have R
n ⊆ A
n , so
c
c
c
that each required value is also an allowable value (but not conversely). More
generally, given c ∈ C, we call s ∈ C an excluded value for c if no product
g
g
g
i∈M c i with i∈M c i = c contains s, that is, in any product i∈M c i whose
value is equal to c, we have c i = s for no i ∈ M . We let E c denote the set of
)
co
all excluded values for c, and let A c denote the complement (E c
of E c and
call it the set of all allowable values for c. Note finally that when confusion
might otherwise result, we will superscript each of the sets introduced above
40 For full details of the sketch we present here, the reader should consult the following
papers: [Seda and Lane, 2005], [Lane and Seda, 2006], [Komendantskaya et al., 2007] and
also [Lane and Seda, 2009].